THIS. No one else has seem to have even brought up, as just a potential idea, a time delay only for the public stream but allow pre-confirmed friends to view the stream live. Or instead of pre-confirmed friends, people who were invited and responded to a Facebook event. Or something.

That would solve the birthday use case while inhibiting the virality.

That's not necessarily good! Maybe the ability to live-streaming police brutality or vote stuffing or whatever, in a way that could go viral, is an important public good. But no one seems to have brought that up, either.